Find the Remainder (x^4-3x^3+4x^2-6x+4)÷(x-2)

$(x^{4} - 3 x^{3} + 4 x^{2} - 6 x + 4) \div (x - 2)$

To calculate the remainder, first divide the polynomials.

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$2$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$6 x$

$4$

Divide the highest order term in the dividend $x^{4}$ by the highest order term in divisor $x$ .

$x^{3}$

$x$

$2$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$6 x$

$4$

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$2$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$6 x$

$4$

$x^{4}$

$2 x^{3}$

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} - 2 x^{3}$

$x^{3}$

$x$

$2$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$6 x$

$4$

$x^{4}$

$2 x^{3}$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$x$

$2$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$6 x$

$4$

$x^{4}$

$2 x^{3}$

$x^{3}$

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$x$

$2$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$6 x$

$4$

$x^{4}$

$2 x^{3}$

$x^{3}$

$4 x^{2}$

Divide the highest order term in the dividend $- x^{3}$ by the highest order term in divisor $x$ .

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$6 x$

$4$

$x^{4}$

$2 x^{3}$

$x^{3}$

$4 x^{2}$

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$6 x$

$4$

$x^{4}$

$2 x^{3}$

$x^{3}$

$4 x^{2}$

$x^{3}$

$2 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $- x^{3} + 2 x^{2}$

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$6 x$

$4$

$x^{4}$

$2 x^{3}$

$x^{3}$

$4 x^{2}$

$x^{3}$

$2 x^{2}$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$6 x$

$4$

$x^{4}$

$2 x^{3}$

$x^{3}$

$4 x^{2}$

$x^{3}$

$2 x^{2}$

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$6 x$

$4$

$x^{4}$

$2 x^{3}$

$x^{3}$

$4 x^{2}$

$x^{3}$

$2 x^{2}$

$6 x$

Divide the highest order term in the dividend $2 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$x^{2}$

$2 x$

$x$

$2$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$6 x$

$4$

$x^{4}$

$2 x^{3}$

$x^{3}$

$4 x^{2}$

$x^{3}$

$2 x^{2}$

$6 x$

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$2 x$

$x$

$2$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$6 x$

$4$

$x^{4}$

$2 x^{3}$

$x^{3}$

$4 x^{2}$

$x^{3}$

$2 x^{2}$

$6 x$

$2 x^{2}$

$4 x$

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{2} - 4 x$

$x^{3}$

$x^{2}$

$2 x$

$x$

$2$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$6 x$

$4$

$x^{4}$

$2 x^{3}$

$x^{3}$

$4 x^{2}$

$x^{3}$

$2 x^{2}$

$6 x$

$2 x^{2}$

$4 x$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$x^{2}$

$2 x$

$x$

$2$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$6 x$

$4$

$x^{4}$

$2 x^{3}$

$x^{3}$

$4 x^{2}$

$x^{3}$

$2 x^{2}$

$6 x$

$2 x^{2}$

$4 x$

$2 x$

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$x^{2}$

$2 x$

$x$

$2$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$6 x$

$4$

$x^{4}$

$2 x^{3}$

$x^{3}$

$4 x^{2}$

$x^{3}$

$2 x^{2}$

$6 x$

$2 x^{2}$

$4 x$

$2 x$

$4$

Divide the highest order term in the dividend $- 2 x$ by the highest order term in divisor $x$ .

$x^{3}$

$x^{2}$

$2 x$

$2$

$x$

$2$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$6 x$

$4$

$x^{4}$

$2 x^{3}$

$x^{3}$

$4 x^{2}$

$x^{3}$

$2 x^{2}$

$6 x$

$2 x^{2}$

$4 x$

$2 x$

$4$

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$2 x$

$2$

$x$

$2$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$6 x$

$4$

$x^{4}$

$2 x^{3}$

$x^{3}$

$4 x^{2}$

$x^{3}$

$2 x^{2}$

$6 x$

$2 x^{2}$

$4 x$

$2 x$

$4$

$2 x$

$4$

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x + 4$

$x^{3}$

$x^{2}$

$2 x$

$2$

$x$

$2$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$6 x$

$4$

$x^{4}$

$2 x^{3}$

$x^{3}$

$4 x^{2}$

$x^{3}$

$2 x^{2}$

$6 x$

$2 x^{2}$

$4 x$

$2 x$

$4$

$2 x$

$4$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$x^{2}$

$2 x$

$2$

$x$

$2$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$6 x$

$4$

$x^{4}$

$2 x^{3}$

$x^{3}$

$4 x^{2}$

$x^{3}$

$2 x^{2}$

$6 x$

$2 x^{2}$

$4 x$

$2 x$

$4$

$2 x$

$4$

$0$

Since the remander is $0$ , the final answer is the quotient.

$x^{3} - x^{2} + 2 x - 2$

Since the final term in the resulting expression is not a fraction, the remainder is $0$ .

$0$

Find the Remainder (x^4-3x^3+4x^2-6x+4)÷(x-2)

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